Graphs and Combinatorics 5, 95-106 (1989) Combinatorics Summary: Graphs and Combinatorics 5, 95-106 (1989) Graphsand Combinatorics © Springer-Verlag 1989 Legitimate Coloringsof ProjectivePlanes N. Alon 1. and Z. FiJredi 2 1 Department of Mathematics, Sackler Faculty of ExactSciences, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel 2 Mathematical Institute of the Hungarian Academy of Sciences, Budapest, P.O.B. 127, H-1364, Hungary Abstract. For a projective plane Pn of order n, let X(Pn)denote the minimum number k, so that there is a coloring of the points of P~in k colors such that no two distinct lines contain precisely the same number' of points of each color. Answering a question of A. Rosa, we show that for all sufficiently large n, 5 < X(Pn)< 8 for every projective plane P, of order n. 1. Introduction Let P = Pn = (P, ~) be a projective plane of order n, with a set of points P and a set of lines ~. As is well known, P has n2 + n + 1 points and n2 + n + 1 lines with n + 1 points on every line. A g-colorin9 of P is a function f from P to the set {1, 2..... X}, which may also be viewed as the (ordered)x-partition (P1, P2 ..... Px) of P defined by Pi = f-l(i). Let C be a X-coloring of P, corresponding to the Collections: Mathematics